The electronic structure of epitaxial graphene—A view from angleresolved photoemission spectroscopy
Abstract and Keywords
This article analyzes the electronic structure of epitaxial graphene using angleresolved photoemission spectroscopy (ARPES). It first describes how the carbon atoms in graphene are arranged before discussing the growth and characterization of graphene samples. It then considers the electronic structure of epitaxial graphene, along with the gap opening in singlelayer epitaxial graphene. It also examines possible mechanisms for the gap opening in graphene, including quantum confinement, mixing of the states between the Brillouin zone corner K points induced by scattering, and hybridization of the valence and conduction bands caused by symmetry breaking in carbon sublattices. Clear deviations from the conical dispersions are observed near the Diracpoint energy, which can be interpreted as a gap opening attributed to graphene–substrate interaction. Graphene–substrate interaction is thus a promising route for engineering the bandgap in graphene.
Keywords: electronic structure, epitaxial graphene, angleresolved photoemission spectroscopy, gap opening, quantum confinement, Brillouin zone, scattering, conduction bands, symmetry breaking, carbon sublattices

14.1 Introduction 441
14.2 Electronic structure of graphene 442
14.3 Sample growth and characterization 445
14.7 Conclusions 460
Acknowledgments 461
References 461
14.1 Introduction
Graphene, the fundamental building block for all graphitic materials, is a twodimensional sheet of carbon atoms arranged in a honeycomb lattice with sp^{2} bonding. The sp^{2} hybridization between one sorbital and two porbitals leads to the formation of σ bonding between carbon atoms, while the π orbitals perpendicular to the plane lead to the formation of the halffilled bands, the πbands. These halffilled bands are the basis for most of the fascinating properties of graphene, and its full potential is still not reached. Although the theoretical study of graphene started in the 1950s (McClure 1957), the experimental study of graphene had not been realized until the recent discovery and characterization of exfoliated graphene by Novoselov et al. (2004) and epitaxial graphene by Berger et al. (2004). Immediately after that, graphene was found to exhibit various intriguing properties unexpected from conventional materials. For example, the charge carriers in graphene are massless Dirac fermions with mobility higher than that of silicon and the doping can be tuned from electrons to holes through a gate voltage (Novoselov et al. 2004). Also, the quantum Hall effect shows halfinteger numbers (Novoselov et al. 2005; Zhang et al. 2005). Because of its fundamental importance in physics as a realization of a relativistic condensedmatter system, as well as its application potentials in nextgeneration electronics, the research interest in graphene has been rising rapidly (Geim et al. 2007; Castro Neto et al. 2008). (p. 442)
14.2 Electronic structure of graphene
The carbon atoms in graphene are arranged in a honeycomb lattice as shown in Fig. 14.1. The unit cell of graphene contains two geometrically different carbon sublattices A and B. The lattice vectors can be written as $\begin{array}{c}{\stackrel{\rightharpoonup}{a}}_{1}=a\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)\\ {\stackrel{\rightharpoonup}{a}}_{1}=a\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)\end{array},$, where a is the carbon–carbon lattice constant √a = 1.42 Å . The corresponding reciprocal lattice vectors are $\begin{array}{c}{\stackrel{\rightharpoonup}{b}}_{1}=\frac{4\pi}{\sqrt{3a}}\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)\\ {\stackrel{\rightharpoonup}{b}}_{2}=\frac{4\pi}{\sqrt{3a}}\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)\end{array}$ and the Brillouin zone also shows a hexagonal shape.
In graphene, the p electrons form the sp^{2} bonding. The intraplanar interaction between the 2s, 2p_{x}, 2p_{y} atomic orbitals form the strongly covalent σ orbitals, which give rise to three bonding σ bands (Fig. 14.2) and three antibonding σ_{∗} bands. The outofplane p_{z} atomic wavefunction forms weakly van der Waals π bonds, which give rise to the π and π^{∗} bands that cross the Fermi energy at the six corners of the hexagonal Brillouin zone. (p. 443)
The dispersion of the π and π^{∗} bands is
where γ∼3 eV (Dresselhaus et al. 1981) is the nearestneighbor hopping integral, $a=\sqrt{3{a}_{0}}$ with the carbon–carbon lattice constant a_{0} = 1.42 Å . The valence and conduction bands touch at the six corners of the Brillouin zone. Since each carbon atom contributes one electron, the valence band is completely filled up to the Fermi level, while the conduction band is empty. Because of the zero separation between the bottom of the conduction band and the top of the valence band, graphene is also known as a semimetal or zerogap semiconductor.
The most peculiar property of graphene is in the region near the Brillouin zone corners where the valence and conduction bands merge. Expanding the dispersions near the Brillouin zone corner K point, $\stackrel{\rightharpoonup}{k}=\stackrel{\rightharpoonup}{K}+\stackrel{\rightharpoonup}{\kappa}$, the dispersion can be written as $E\left(k\right)=\pm \frac{\sqrt{3a}}{2}\gamma \left\stackrel{\rightharpoonup}{\kappa}\right=\pm \hslash {v}_{F}\kappa $, where ${v}_{F}=\frac{\sqrt{3}a\gamma}{2\hslash}$. This dispersion relation is analogous to that of relativistic particles $E=\pm \sqrt{{m}^{2}{c}^{4}+{c}^{2}{p}^{2}}$ with zero effective mass m = 0, and the speed of light c is replaced by the Fermi velocity ћvF, which is approximately 300 times smaller. Therefore, electrons in graphene are governed by a twodimensional version of the relativistic theory introduced by Dirac. Because of this similarity, the lowenergy electrons in graphene are also described as massless “Dirac fermions” and the points where the valence and conduction bands merge are termed “Dirac points”.
Figure 14.3 shows a schematic drawing of the lowenergy dispersions near E_{F}. The peculiar linear dispersion near E_{F} results in many intriguing properties. First, the “Fermi surface” in graphene contains only six points, different from a real surface common in metals. Because of the finite number of points at the Fermi surface, the density of states vanishes at E_{F}. Moreover, electrons in graphene travel with a constant velocity, in contrast to electrons in a semiconductor, where the dispersion shows a quadratic behavior near the top and bottom of the bands. An electron in the latter case is modelled as a quasiparticle with a finite (“massive”) effective mass ${m}^{*}={\hslash}^{2}{\left(\frac{{d}^{2}E}{d{k}^{2}}\right)}^{1}$ and can be described by nonrelativistic theory formulated in Schrödinger’s equation. The effective mass m^{∗} is usually different from the noninteracting electron mass and this mass renormalization is used to take into account the effects of electron–electron and electron–phonon interactions.
(p. 444) The completely linear dispersion near E_{F} and the massless Dirac fermions are expected in singlelayer graphene only in the case of a perfect crystal. In the presence of additional interactions, e.g. breaking of the carbonsublattice symmetry, the valence and conduction bands will hybridize; causing a finite gap between the valence and conduction bands. One extreme case is BN, which has a similar hexagonal structure and two completely different sublattices, which results in a gap of up to a few eV (Blase et al. 1995). Similar gap opening can be expected if the potentials on the two carbon sublattices are inequivalent. When a gap exists between the valence and conduction bands, the Dirac fermions will acquire a finite mass. However, despite a small deviation near E_{F} from the linear dispersion, the overall dispersion will still show a linear behavior when moving far enough away from the Dirac point.
In the case of bilayer and trilayer graphene, the number of π bands will increase with the number of graphene layers. The overall linear dispersion is still preserved, along with some additional parabolic perturbations near E_{F} (Fig. 14.4; Partoens and Peeters 2007). When the number of graphene layers exceeds 10, the electronic structure of multilayer graphene is basically that of graphite.
(p. 445) 14.3 Sample growth and characterization
Graphene research has focused mainly on two types of samples, exfoliated graphene and epitaxial graphene. The exfoliated graphene sample is produced by mechanical exfoliation of graphite flakes followed by deposition on a Si wafer coated with 300 nm of SiO_{2} (Novoselov et al. 2004, 2005). Exfoliated graphene samples have been widely used in transport measurements as they have very high mobility and its structure makes it easy to apply a gate voltage to tune the charge carriers. However, exfoliated graphene samples have low yield and they are typically small (of the order of 10 μm, much smaller than the size of the synchrotron beam ∼100 μm used in ARPES) and therefore it is difficult to study the electronic structure of exfoliated graphene using ARPES. So far, the electronic structure of graphene has been mostly reported only in the other type of graphene sample, epitaxial graphene. It is well known that thick graphite samples can be grown by cracking of hydrocarbon gas on various metals (McConville et al. 1986; Land et al. 1992) or by thermal decomposition of carbide wafers (Van Bommel et al. 1975; Nagashima et al. 1993). However, it was not until recent years that the singlelayer epitaxial graphene sample on SiC was fully characterized (Berger et al. 2004). The advantage of epitaxial graphene is that the samples can be made much larger (mm scale) and therefore they are suitable for ARPES study to extract the electronic structure directly. Technologically, epitaxial graphene is important since the existing Si technology can be incorporated to mass produce epitaxial graphene samples (Berger et al. 2006; de Heer et al. 2007; Kedzierski et al. 2008).
Graphene samples on SiC wafer can be grown by thermal decomposition of the SiC wafer. The choice of SiC substrate has the advantage that SiC has a bandgap of ∼2.6 eV and therefore its electronic structure does not interfere with that of graphene near E_{F}. There are two terminations for the SiC wafers, Siterminated and Cterminated. The graphene on the Cterminated 4HSiC shows more inplane orientational disorder (Hass et al. 2006) than graphene on Siterminated 6HSiC. Therefore, graphene on Siterminated face is more favorable for ARPES studies. The 6HSiC is doped with nitrogen to have a resistivity of ∼0.2Ω cm^{−1} unless mentioned otherwise. The growth process is monitored with lowenergy electron diffraction (LEED) or lowenergy electron microscopy (LEEM). The wafer was cut into pieces of a few millimeters in size by using a diamond saw, and cleaned with acetone and isopropyl alcohol in an ultrasonic bath. The clean wafers were then mounted to a Ta sample holder and loaded into an ultrahighvacuum preparation chamber with base pressure 1 × 10^{−10} torr. The wafer was then annealed at 850°C under silicon flux with deposition rate about 3 Å per minute to remove the native surface oxides by the formation of volatile SiO. Figure 14.5 shows the LEED pattern at different stages during the growth process. After the initial cleaning under Si flux, the LEED pattern shows a 3 × 3 reconstruction with respect to the SiC substrate, as has been well documented in the literature (Kaplan 1989; Bermudez et al. 1995; Forbeaux et al. 1998; Starke 1998). A subsequent annealing at 1000°C in the absence of Si flux shows a 1 × 1 pattern. More annealing at 1100°C shows the $\left(\sqrt{3}\times \sqrt{3}\right)R{30}^{\circ}$ reconstruction. (p. 446)
This corresponds to a 1/3 layer of Si adatoms in threefold symmetric sites on top of the outermost SiC layer (Northrup 1995; Owman 1995). Further annealing at 1250°C results in the $\left(6\sqrt{3}\times 6\sqrt{3}\right)R{30}^{\circ}$ pattern, which indicates the formation of the carbonrich layer. This carbonrich layer is formed by carbon atoms arranged in the same structure as graphene. However, this layer forms only the σ bands but no π bands characteristic of singlelayer graphene (Emtsev et al. 2007). Therefore, this carbonrich layer is called a buffer layer (Emtsev et al. 2007; Mattausch et al. 2007; Varchon et al. 2007). Finally, annealing at 1400°C forms the graphene layer on top of the buffer layer (Fig. 14.6).
Even though LEED is a good indicator for the different stages of the growth, the distinction between graphene and the carbonrich (buffer layer) is not obvious, and LEED is not sensitive to the graphene thickness. Therefore, the sample thickness needs to be characterized using other methods.
Lowenergy electron microscopy is a powerful technique to study the surface topography and the dynamics of the growth process (Bauer 1994; Phaneuf et al. 2003), as well as characterizing the sample thickness. LEEM is a surfacesensitive imaging tool that collects the backscattered electrons ranging from 1 to 100 eV. The low energy of the electrons used distinguishes LEEM from other electron microscopies, e.g. transmission electron microscopy (TEM) where the electron energy is typically 100 000 eV. The use of lowenergy electrons has the advantage that a large fraction of electrons will be backscattered elastically and that the backscattered electrons in this lowenergy range are very sensitive to the physical and chemical properties of the surface. Thus, depending on the energy of the incident electrons, subtle differences in the local atomic structure or composition can result in dramatic reflectivity contrast in LEEM. Another advantage of LEEM over other microscopy is that the images can be taken instantly, allowing realtime study of the dynamics. For example, Fig. 14.7 shows the LEEM image before cleaning the SiC wafer at 850°C under Si flux. The LEEM image shows that after the initial cleaning, the sample surface becomes smoother. The direct visualization of the growth process allows a fine tuning of the growth parameters to achieve highquality samples, and to obtain graphene samples with various characteristics, e.g. different graphene terrace size.
In addition to studying the surface topography, LEEM can also yield direct information about the graphene sample thickness by studying the quantumwell states on a substrate (Hibino et al. 2007; Ohta et al. 2008). The coherent reflected electrons from the surface and the interface between the graphene and the substrate will interfere with each other and form interference patterns in the reflectivity as a function of electron energy. The number of quantum oscillations directly reflects the number of states, or the thickness of the graphene sample, and can therefore determine the sample thickness. Figure 14.8 shows a LEEM image with intensity ranging from white, gray and black. The energy scan shows zero, one and two minima in the energy scan, allowing us to determine these regions to be buffer layer, single layer and bilayer graphene, respectively. Therefore, the LEEM image shows that the epitaxial graphene has regions of single and bilayer graphene, as well as the buffer layer. The singlelayer graphene is the majority and the terrace size of the singlelayer graphene can be controlled by tuning the annealing temperature and annealing time.
Additional ways to characterize the sample thickness are: Auger electron spectroscopy (Berger et al. 2004), Xray photoelectron spectroscopy (Rolling et al. 2006) and angleresolved photoemission spectroscopy (Ohta et al. 2006).
14.4 Electronic structure of epitaxial graphene
Angleresolved photoelectron spectroscopy (ARPES) is a direct and powerful tool to probe the electronic structure of a material (Damascelli et al. 2003). The basic principle behind this technique is the photoelectric effect. When a beam of light with photon energy hν shines on a clean sample surface, photoelectrons will be emitted from the sample, which are collected by an electron analyzer as a function of kinetic energy E_{k} and angle θ (Fig. 14.9). The energy is conserved in the photoemission process, therefore from the kinetic energy E_{k} of photoelectrons and the work function φ, the binding energy of electrons inside the solid can be determined as E_{B} = hv − φ − E_{k}. For singlecrystal samples, because of the translational symmetry in the crystal plane and the negligible wave vector of the photons, the inplane momentum p// is preserved. Therefore, the electron binding energy and inplane momentum can be determined as $\begin{array}{ll}{E}_{B}\hfill & =hv\varphi {E}_{k}\hfill \\ {p}_{//}\hfill & =\hslash {k}_{//}=\sqrt{2m{E}_{k}}\mathrm{sin}\theta \hfill \end{array}$. Therefore, ARPES can directly measure the dispersion E(k) and map out the Fermi surface. Moreover, ARPES directly measures the singleparticle spectral function A(k, ω), which not only contains the band dispersion, but also the selfenergy due to different interactions. (p. 448)
Figure 14.10 shows ARPES data taken in singlelayer epitaxial graphene along the highsymmetry direction K in a large energy range. The π bands at low binding energy and the σ bands at high binding energy can be clearly observed, similar to the calculated dispersion shown in Fig. 14.2. In addition to the π and σ bands, the bands from the SiC substrate (pointed to by the arrow in panel (b)) can also be observed and the intensity of these bands increases with higher photon energy. The advantage of using semiconducting SiC as substrate is that the SiC bands are at much higher binding energy, and therefore the lowenergy π band near E_{F}, which is the main focus of this study, can be easily separated from the substrate.
Figure 14.11 shows the intensity map measured at E_{F}, −0.4 and −1.2 eV taken on singlelayer epitaxial graphene near the Brillouin zone corner K. At the E_{F} (panel (a)), the intensity map shows a small circular pocket instead of a single point as expected, suggesting that the sample is doped. This pocket decreases in size when going down in energy and it appears to be a point near −0.4 eV. Beyond −0.4 eV, the size of the pocket increases again and it shows an overall circular behavior. The modulation of the intensity inside and outside the first Brillouin zone is known to be caused by the dipole matrix element in ARPES (Shirley et al. 1995). These constant energyintensity maps are in overall agreement with the conical dispersion. However, the sample is electron doped and the Dirac point energy E_{D} lies below the Fermi energy E_{F}.
The conical dispersion and the electron doping can also be studied from the dispersion. Figure 14.12 shows the dispersion along a line through the K point. Two cones dispersing in opposite directions, one upward and another downward, can be clearly observed. The center point between these two cones, which is defined as the Dirac point energy E_{D}, lies at −0.4 eV. Figure 14.12(b) shows the momentum distribution curves (MDCs), intensity as a function of momentum, for energies between E_{F} and −1.5 eV. The dispersion can be extracted by following the peak positions in the MDCs, which shows an almost linear behavior for both the upper cone and lower cone. Overall, the data presented so far on singlelayer epitaxial graphene show that the dispersion is in agreement with the conical dispersion and that the sample is electron doped.
Figure 14.13 shows the evolution of the dispersion with graphene sample thickness. From singlelayer to bilayer graphene, the π band splits into two bands as a result of interlayer interaction. In addition, the bilayer graphene sample is also electron doped and there is a finite electron pocket at E_{F}. For trilayer graphene, E_{D} moves even closer to E_{F}, and there is still a finite electron pocket at E_{F} (Ohta et al. 2007; Zhou et al. 2007). Eventually, when the sample thickness is infinite, ED lies almost at E_{F} (Zhou et al. 2006).
The thickness dependence in Fig. 14.13 suggests that the electron doping comes from the interface. To test if the doping is related to the doping of the SiC substrate or the Si atoms at the interface, we measured two bilayer graphene samples on a different substrate 4HSiC with resistivity of 105Ωcm^{−1} compared to 0.2Ωcm^{−1} in the 6HSiC substrate studied previously. (p. 450)
Figure 14.14 shows that even though the resistivity of the substrate is different by a factor of 10^{5}, E_{D} is still at the same energy of −0.29 eV within an error bar of 20 meV. Figure 14.15 summarized the data taken on graphene samples on both 6HSiC and 4HSiC. It is clear tat even though E_{D} changes with graphene sample thickness, it is independent of the doping of the SiC substrate. This comparison shows that the doping of the graphene sample is related to the charge transfer at the interface from the Si atoms (Zhou et al. 2007), and it not determined by the doping of the substrate. This is in agreement with the thickness dependence, since for the thicker graphene sample, it is farther away from the interface, and therefore the change in E_{D} is smaller and E_{D} is closer to E_{F}.
Data presented so far show that epitaxial graphene samples are electron doped. The electron doping is most likely associated with the charge transfer from the Si atoms at the interface and not related to the doping of the substrate. The dispersions show an overall conical behavior. However, near the Dirac point energy (Fig. 14.13(a)), there is an anomalous vertical region. To understand this vertical region near E_{D}, a more detailed analysis of the dispersion near E_{D} is required.
(p. 451) 14.5 Gap opening in singlelayer epitaxial graphene
Figure 14.16 shows a more detailed analysis of the data near E_{D} at the Brillouin zone corner K. The dispersions can be extracted from the energydistribution curves (EDCs), intensity scans as a function of energy for fixed momentum. In the EDCs, there are two peaks that correspond to the valence and conduction bands. It is interesting to note that even at the K point, where these two bands are expected to meet for a perfect conical dispersion, there are still two peaks in the EDCs. This presence of two peaks at the K point suggests that the valence and conduction bands do not merge at the K point. Instead, there is a finite separation between these two bands. To extract the dispersion for both the valence and conduction bands, we fit EDCs with two Lorentzians multiplied by the Fermi–Dirac function. The extracted dispersions are shown as white lines in panel (a), which clearly show a finite separation of ∼0.26 eV between the valence and conduction bands. This anomalous region between the valence and conduction bands can also be observed in the MDCs shown in panel (c). Away from E_{D}, two peaks are clearly observed in the MDCs, while in an extended region between −0.3 eV and −0.5 eV, only one peak is observed in the MDC and the peak position does not change with energy. The nondispersive MDC peak is a typical ARPES feature of the gap (Shen et al. 2005). Panel (d) shows the angleintegrated intensity curve, which is proportional to the density of states. At high binding energy, the angleintegrated intensity curve shows a linear behavior, while near the Dirac point energy, there is a dip suggesting suppression of the density of states below the expected linear behavior. This is in agreement with the gap opening near E_{D}. In summary, Fig. 14.16 shows that the dispersion near E_{D} deviates from the linear behavior and there is a finite gap opening between the valence and conduction bands. One interesting observation is that there is still finite intensity inside the gap region.
Figure 14.17 shows how the dispersions evolve across the K point. Panels (a,b) show the intensity maps at E_{F} (above E_{D}) and −1 eV (below E_{D}), respectively. Above E_{D}, the intensity is stronger outside the first Brillouin zone, while below E_{D} the intensity is stronger inside the first Brillouin zone. This intensity modulation is related to the dipole matrix element in graphene (Shirley 1995) and can be utilized to identify the exact K point. Panels (c–h) show the data taken before and after crossing the K point, with a very fine step of 0.05 deg close to the K point, which corresponds to a step of 0.0026 Å^{−1} to make sure to catch the K point as accurately as possible. Before crossing the K point (panel (c)), the data shows a stronger intensity for the valence band, while after crossing the K point (panels (g,h)), the data show a stronger intensity for the conduction bands, in agreement with the intensity modulation in panels (a,b). In all the panels, the valence and conduction bands can be followed from the EDC peak positions. The most important information comes from the EDC at k_{y} = 0, where the separation between these two bands is the smallest in each panel. Panel (i) shows the EDCs at k_{y} = 0. Two peaks can be identified in the EDCs, and the intensity modulation between these two peaks switches before and after crossing the K point. This intensity modulation enables us to identify that the K point is for cut (e), since for this cut the EDC shows an almost identical intensity for the two peaks and the gap is smallest. The gaps for the various cuts taken can be plotted a function of the momentum k_{x}, as shown in panel (j). The consistent trend of the gap with the minimum value of ∼0.26 eV at the K point shows that the gap is not an artifact due to sample misalignment, therefore the upper cone and lower cones are separated by a finite amount and there is a bandgap of ∼0.26 eV in singlelayer graphene. In addition, a finite intensity is always observed inside the gap region for all the cuts.
The dispersions near the K point can also be studied by measuring along another geometry (see Fig. 14.18(a)), which shows asymmetric intensity for the dispersions on the two sides of the K point. This geometry has the advantage of showing only one branch of the dispersion, thereby enabling an easier fit of the MDCs to extract the dispersion and the MDC width, with the additional disadvantage that the Dirac point cannot be envisioned directly. In addition, one caveat of the MDC analysis is that the presence of a peak does not necessarily mean the presence of a quasiparticle, therefore one also needs to be careful about the interpretation of the MDC dispersions (Zhou et al. 2008b). Panels (b) and (c) show two cuts, one through the K point and another one off the K point. In panel (b), the top of the valence band and the bottom of the conduction band for the cut through the K point (panel (b)) are determined from the symmetric geometry discussed previously and are labelled by the two gray arrows in panel (b). The black arrows in panel (c) label the top and bottom of the bands determined for the cut off the K point. The data show an overall similarity between these two cuts with a suppression of intensity in the region near the Dirac point energy. The extracted MDC dispersions show an even larger deviation from the linear dispersion for the cut off the K point. In addition, the MDC width shows an anomalous region near E_{D}, where multiple peak structures appear. This anomalous region has been discussed previously by Bostwick et al. (2006) and was attributed to electron–plasmon interaction. However, it is interesting to note that energies for the top of the valence band and bottom of the conduction band coincide with the two peak positions in the anomalous region for both cuts (b) and (c), and the amplitude of the peaks is larger for the cut off the K point. This suggests that the anomalous region near E_{D} has a similar origin for the cuts through and off the K point (Zhou et al. 2008b). In the latter case, it is known from the conical dispersion that a gap is definitely expected. This is in agreement with previous EDC analysis of Fig. 14.16 and Fig. 14.17, where a gap is present.
Figure 14.19 shows that the anomalous peaks in the MDC width near E_{D} can be explained by the gap (Zhou et al. 2008b). Panels (a) and (b) show the simulated data with the dipole matrix element that enhances one branch of the dispersion, similar to the case in graphene. Applying a similar MDC fit and extracting the MDC width, the MDC width shows two peaks in the gap region and the peak is larger for the larger gap. This similarity with respect to the experimental data discussed in Fig. 14.18 shows that the gap opening is the most likely origin for explaining the deviation near E_{D} and the anomalous peaks in the MDC width, and not due to electron–plasmon interaction. (p. 454)
14.6 Possible mechanisms for the gap opening
There are a few possible mechanisms that can open up a gap in graphene: quantum confinement (Nakada et al. 1996; Brey et al. 2006; Son et al. 2006; Han et al. 2007; Nils et al. 2007; Nilsson et al. 2007), mixing of the states between the K and Kʹ points induced by scattering, and hybridization of the valence and conduction bands caused by breaking of carbon sublattice symmetry (Manes et al. 2007). We will examine all of these possibilities and we think that the most likely scenario is the breaking of the carbonsublattice symmetry.
(p. 455) 14.6.1 Quantum confinement
The gap induced by quantum confinement is well known in the case of carbon nanotubes (Saito et al. 1992) and has been recently extended to graphene nonoribbons (Nakada et al. 1996; Son et al. 2006; Han et al. 2007; Trauzettel et al. 2007). Carbon nanotube is a sheet of rolled graphene, therefore additional boundary conditions apply to carbon nanotube. When the graphene sheet is rolled along a certain direction to form carbon nanotubes, the electronic states will be quantized along this direction (Fig. 14.20). For certain directions where the bands do not cross the Dirac point, the carbon nanotube will be a semiconductor with a finite size gap (Saito et al. 1992). The electronic states near E_{F} are determined by the intersection of the allowed K points with the dispersion cones at the K point. The size of the gap increases as the carbon nanotube becomes smaller, and scales inversely with the width of the ribbons. The dependence of the gap size on the graphene ribbon width has been recently shown by Han et al. in exfoliated graphene (Han et al. 2007). It is important to note that for ribbons as small as 10 nm, the gap size can be as big as 200 meV, while for ribbons larger than 30 nm, the gap would be smaller than 10 meV. Therefore, it is important to measure the gap in our graphene samples with various graphene ribbon sizes to test whether the scenario of quantum confinement can account for the large gap observed here.
Singlelayer graphene samples with various terrace size can be obtained by controlling annealing temperature and annealing time. The realtime observation of the growth process is a unique advantage to control the ideal growth conditions and obtain graphene samples with various sizes. As discussed previously in Section 14.2, LEEM is a powerful tool to study the surface topography and identify the singlelayer graphene regions. Therefore, the graphene terrace size can be directly measured from the LEEM image. Figure 14.21 shows a LEEM image with a majority of singlelayer graphene with smaller percentage of bilayer graphene and buffer layer. The graphene terrace size and the error bar is quantified by taking the average and the standard deviation of the graphene terrace sizes crossed by the two lines along the diagonal direction.
Figure 14.22 shows the LEEM images for a few graphene samples with various singlelayer graphene terrace sizes. The samples contain a majority of singlelayer graphene with different terrace sizes, along with a much smaller fraction of bilayer graphene and buffer layer. The singlelayer graphene regions are the focus here. The singlelayer graphene terrace size in each panel is characterized using the method described above. In these samples, with increasing annealing, the graphene terrace size can be controlled from 50 nm (panel (e)) to 180 nm (panel (a)). After the LEEM measurements, the same samples were transported to the ARPES chamber. After annealing at 700°C to remove the gas adsorbed on the graphene surface, the samples were measured with ARPES to yield direct information about the electronic structure. Figure 14.23 shows the corresponding ARPES data for the same samples measured in the same geometry. Following the same procedure, the gap can be determined by measuring the distance between the valence band and conduction band extracted from the EDC peak positions. When the graphene terrace size decreases from panel a to panel e, the gap increases slightly. (p. 456)
Figure 14.24 shows the direct correlation of the gap size measured from ARPES with the graphene terrace size obtained from the LEEM measurements (Zhou et al. 2008a). It is clear that the gap slightly increases for the smallest graphene terrace size. However, the gap does not change within the experimental uncertainty even when the graphene terrace size exceeds 150 nm. This is in contrast to the gap induced by quantum confinement in the exfoliated graphene nanoribbons where the gap is below 10 meV when the graphene terrace size is larger than 30 nm. This direct comparison shows that even though quantum confinement can contribute to enhance the gap size, quantum confinement cannot explain the large gap of 180 meV still observed in largest graphene terrace size. Therefore, quantum confinement is not the main mechanism for the gap opening in epitaxial graphene (Zhou et al. 2008a).
14.6.2 InterDiracpoint scattering
Another possible way to open up a gap is to hybridize the electronic states at K and Kʹ (Manes et al. 2007) as schematically shown in Fig. 14.25. This requires breaking of the translational symmetry. It is known that in graphene there are reconstructions of 6 × 6 and (6✓3 × 6✓3)R30° (Tsai et al. 1992; Forbeaux et al. 1998). However, the scattering vectors related with these two constructions are much smaller than the K–K^{ʹ} distance, which is required to mix the states at K and Kʹ. Higherorder scattering process involving consecutive small scattering vectors is weak in general and is an unlikely source for the gap opening. Impurity scattering can also mix the states at K and K^{ʹ} (Rutter et al. 2007). However, this would give rise to a gap that strongly depends on impurity concentration. This is in contrast to our findings, where the gap is similar for all the samples studied.
14.6.3 Breaking of the carbonsublattice symmetry
The most likely scenario to explain the gap that we observed in epitaxial graphene is the breaking of the carbon sublattices (see Fig. 14.26). We note that a perfectly conical dispersion is expected only in the case of perfect singlelayer epitaxial graphene, where the potentials on the two carbon sublattices are the same. When this symmetry breaks down, the hybridization between the valence and conduction bands can open up a gap. This has been well known in the extreme case of BN, which has a similar honeycomb structure and completely different sublattices in the unit cell. In the case of BN, the bandgap can be as big as 5.8 eV (Blase et al. 1995).
In the case of singlelayer epitaxial graphene, since the sample is grown on top of the SiC substrate and there is a buffer layer between graphene and the SiC substrate, the buffer layer can break the symmetry between the two carbon sublattices. A prediction of this scenario is that breaking of the sixfold rotational symmetry of graphene near the Dirac point energy. Figure 14.27(e) shows the calculated intensity map at E_{D} using a tightbinding model in the
extreme case when only one of the two carbon sublattices have nonzero potential. The potential modulation imposed by the (6✓3 × 6✓3)R30° reconstruction has been added as a perturbation to the Hamiltonian (Zhou et al. 2007). For energy well above or below E_{D}, the symmetry is restored. Figures 14.27(a–d) show the constant energyintensity maps measured at E_{F}, E_{D} and below E_{D}. The dominant feature in all these intensity maps is the small pockets centered at the K points. In addition, there are six weaker replicas surrounding each K point. The intensity of these replicas is only ∼4% that of the main peak at the K point. These replicas are associated with the (6✓3 × 6✓3)R30° observed in LEED. At E_{F} (panel (a)), the six replicas show similar intensity, while near E_{D} (panel (b)), three of the six replicas (pointed to by the arrows) are enhanced, suggesting that the sixfold symmetry is broken near E_{D}. At higher binding energy far away from the Dirac energy (panel (d)), the six replicas show similar intensity again, showing the restoration of the sixfold symmetry. Our experimental data shown in panels (a–d) are in good agreement with theoretical predictions that show the breaking of the sixfold symmetry near E_{D}. We note that scanning tunnelling microscopy STM measurements (Brar et al. 2007) on epitaxial singlelayer graphene did not report the evidence
of sixfold symmetry breaking. However, the STM data are taken far away from the Dirac point, where it is known that the effect of the symmetry breaking is much weaker and difficult to observe. Therefore, the lack of evidence of sixfold symmetry breaking in STM studies does not contradict our ARPES results.
In addition, there are some interesting structures in the intensity maps near E_{D}. In particular, we observed hexagonal patterns where the intensity is enhanced (see Fig. 14.27(b)). There are two observations associated with these additional patterns. First, the center midsized hexagon around Γ (gray dotted line in panel (b)) almost overlaps with the first Brillouin zone of SiC. Second, all other hexagons (gray broken lines in panel (b)) are not regular, i.e. the six sides that form the hexagon do not have the same length. However, they all pass through the K points. These two observations suggest that the states associated with the hexagonal pattern are related to the buffer layer between the SiC substrate and the graphene.
Figure 14.28 shows a more careful characterization of the ingap states. Figure 14.28(b) shows the MDCs at E_{D} for momenta from k_{x1} to k_{x8}. Three nondispersive peaks are observed in the MDCs, which correspond to the three horizontal segments of the additional hexagonal patterns. This shows that the midgap states have maxima along the hexagonal patterns. Panel (c) shows the EDCs taken at various points in the Brillouin zone. At the K point, the EDC shows two peaks separated by a finite amount, which correspond to the top of the valence band and the bottom of the conduction band, in agreement with the gap. Along the hexagonal pattern, the EDCs show a very broad peak from E_{F} to −0.8 eV, while away from the hexagonal pattern (see, e.g. EDC at the point), the broad peak in the EDC can still be observed. However, the intensity is only half of that along the additional hexagons. The analysis of Fig. 14.28 shows that there are additional states that are peaked at the Diracpoint energy, and these midgap states form interesting hexagonal pattern in the momentum space. The exact origin of these midgap states still requires further studies.
(p. 460) The proposed mechanism of breaking carbonsublattice symmetry to explain the gap opening in epitaxial graphene is also supported by recent abinitio calculation (Kim et al. 2008), which shows a gap near E_{D} with a gap value similar to ARPES measurements. The calculation shows that the graphene layer follows the atomic structure of the buffer layer underneath and there is a corrugation of the height of the graphene layer, which results in a 140meV average potential difference in the two carbon sublattices. Moreover, midgap states are also predicted, which are caused by the interlayer coupling between the π states in graphene and the localized π states in the buffer layer. This is in agreement with the observation of anomalous intensity in the gap region. However, further studies are still needed to understand why the midgap states form such hexagonal patterns in the momentum space as discussed above.
Our proposed mechanism of graphene–substrate interaction as the origin for the opening of the bandgap has important implications, since engineering the bandgap in graphene is an important topic for its applications. It was predicted that when graphene is grown on BN substrate, a gap can also be expected and the gap depends on the orientation of graphene relative to the BN substrate as well as the distance between graphene and the substrate (Giovannetti et al. 2007). If one can tailor graphene–substrate interaction by growing graphene on different substrates, it is possible to engineer the bandgap of graphene in a wider range. Further study to move the Fermi energy inside the gap region with hole dopants to eventually make graphene a semiconductor is also an important topic.
14.7 Conclusions
In summary, we have studied the electronic structure of epitaxial graphene. Clear deviations from the conical dispersions are observed near the Diracpoint energy, and interpreted as a gap opening due to graphene–substrate interaction. This points to graphene–substrate interaction as a promising route to engineer the bandgap in graphene. Even though it might take a long time before graphene’s full application potentials can be fully realized, graphene is definitely a very intriguing system and there is still a lot more to be explored.
(p. 461) Acknowledgments
We thank A.V. Fedorov, D.A. Siegel, G.H. Gweon, F. El Gabaly, A.K. Schmid, K.F. McCarty and J. Graf for experimental assistance, P.N. First, W.A. De Heer for helping with the graphene samples in the initial stage of this project, D.H. Lee, F. Guinea, and A.H. Castro Neto for useful discussions.
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